Ball-Evans approximation problem: recent progress and open problems

TL;DR

The article surveys the Ball-Evans approximation problem, connecting the approximation of Sobolev homeomorphisms by diffeomorphisms or piecewise affine maps to nonlinear elasticity via the energy functional $I(f)=\int_\Omega W(Df(x))\,dx$. It reviews strong planar results, where $f\in W^{1,p}$ can be approximated by smooth diffeomorphisms (and piecewise affine maps) with convergence in $W^{1,p}$ (and uniform convergence in some cases), and it highlights the central open question in dimensions $n\ge 3$. It explains that while higher-dimensional approximability is unresolved, there are counterexamples for small $p$ that obstruct such approximations, pointing to fundamental dimension- and exponent-dependent barriers. The discussion extends to related issues such as incompressible deformations and higher-gradient functionals, underscoring potential impacts on regularity theory for energy minimizers and numerical finite-element strategies. Overall, the paper outlines a roadmap for resolving the higher-dimensional Ball-Evans problem and clarifies its significance for both analysis and applied elasticity.

Abstract

In this paper we give a short overview about the Ball-Evans approximation problem, i.e. about the approximation of Sobolev homeomorphism by a sequence of diffeomorphisms (or piecewise affine homeomorphisms) and we recall the motivation for this problem. We show some recent planar results and counterexamples in higher dimension and we give a number of open problems connected to this problem and related fields.